General Relativity Question (with diagram)

Does the stress-energy tensor depend on direction of the relative velocity of two celestial bodies? Assume vy is directed parallel to the gravitational field of the planet, vx and vz are perpendicular to the field, and that the speed would be the same whichever direction it is in. Does it matter whether the velocity is in the x, y, or z direction?

To oversimplify it a great deal, one can regard the stress-energy tensor as giving the amount of energy and momentum stored in a unit volume. . So in empty space, the stress-energy tensor is zero. If you have matter present (a cloud of gas, a blob of fluid, a block of substance, or a planet) the stress energy tensor will be non-zero. Electromagnetic radiation (such as light) can also contribute to the stress-energy tensor, along with matter.

The interesting thing is that if you know all the components of the stress-energy tensor in one basis (you can think of this as "frame" if you're dealing with special relativity), you can compute the components in any basis or frame you choose.

There is an ambiguity in the concepts here - one can regard the stress-energy tensor as a set of components, and these components change when you change basis vectors (or frames). But one can regard it as representing a physical entity. In the later case, the description of this entity changes depends on the viewpoint - i.e. the choice of basis or frame. But the entity itself is regarded as being "the same entity", one regards the description of the entity as changing but not the entity itself.

Without going into all the details needed for a full understanding, I'll just point out that if you have a spherical baseball, and you view it from a different frame moving at relativistic velocity, Lorentz contraction makes the baseball non-spherical and shrinks it in the direction of motion, which affects it's volume. The stress-energy tensor is needed to have a coherent explanation of the concept of "density" given the Lorentz transform, which changes the shape and volume of the baseball. (It does other things, too, but I won't get into those). So the concept of "a unit volume" that I glossed over hides some tricky details that I'm not attempting to explain at this point, it would get too long and advanced to do a proper explanation.